The so-called Eagle Rock (also known as Hunter's Craig) is located on the shore of the River Forth, West of the River Almond at Cramond, on the Dalmeny Estate. The carving on the rock is on the East side of a small outcrop of rock with a small cave on the West side. It is situated in a rectangular recess cut into the rock measuring approximately 3' x 2' and about 8" deep, some 10' above ground level. A further recess is cut into the rock below the sculpture's frame, which currently houses a Historic Scotland plaque.

It is often said to be that of a Roman imperial eagle, probably carved by troops garrisoned at the nearby Roman fort at Cramond. This idea seems to have been first put forward by J.P. Wood in 1794, who described it as:
This theory was also repeated by W.L. Colvin in the second Statistical Account of Scotland (1845):
The same theory is also repeated in general local history texts today. Even the current Historic Scotland website gives this interpretation, albeit more cautiously: "A much-defaced carving on natural rock said to represent an eagle."
However, on actually viewing the site, the eagle interpretation, let alone a simple Roman provenance seems highly unlikely. For a start, the sculpture is asymmetrical with regard to what remains of and what would have been its limbs, which would rule out a Roman eagle with wings either open or folded. Indeed, it appears to have its right arm pointing downwards, possibly holding something which trails to the floor, while its left arm is folded, apparently holding a club or sceptre-like object over the shoulder:

The actual impression is more of an altar to a pagan deity. The question of which deity and thus the cultural provenance of the carving is then opened. According to RCAHMS in 1929, the deity represented is possibly that of Mercury, and thus once more Roman. But for Anne Ross, the sculpture was "apparently adorned with a horned human head",[3] and from this she uses it as evidence to argue for the existence of a local cult of Cernunnos (the horned god), suggesting a more local, specifically Celtic provenance.

Certainly, the head, such as we can see today, seems large and squarish, opening up the possibility of it having originally been horned. Significantly, no early commentators seem to comment on these alleged horns, but it is possible that their remains lead to the impression of an eagle's ears, and hence, the assumption of a Roman eagle. Unfortunately, though, the head is now so badly worn that the issue of Cernunnos-like horns is probably unresolvable.
One other factor which may be of use in interpreting which deity is represented here is a generally uncommented fact - that there is a small cave to the rear (West) of the carving, on the opposite side of the same rock:

Is it possible that this cave, like many others within the pagan world, was given some significance as an entrance to the Otherworld? If so, it may be that the deity represented on Eagle Rock could have some specific role as a guardian of the Otherworld, or the passage between it and the everyday world.
What is more certain is that the figure appears to be carrying a club or sceptre-like object over its left shoulder. By itself, this would suggest either a Roman Hercules figure, or a Celtic Ogmios figure - two gods which, in Romano-Celtic times, were often associated with each other anyhow. It should also be remarked that Mercury, the interpretation suggested by RCAHMS, was also occasionally associated with Ogmios.
However, regardless of whether we have a shrine here to the Roman gods Mercury or Hercules, or the Celtic gods Cernunnos or Ogmios, the style of the carving and its setting within a rectangular framed inset, specifically suggests the Romano-Celtic period, rather than an unproblematic Roman or Celtic shrine. This would, then, date it to the period of Roman occupation at nearby Cramond, and allow us to keep the possibility of it being executed by the Cramond "Romans". What should be borne in mind here, though, is that many cohorts of "Roman" troops were in fact native Celts themselves, often from Gaul. Occasionally, we also find evidence of them "importing" their Gaulish Celtic deities into Britain, as at the Roman fort of Olicana (modern Ilkley in West Yorkshire, England). Here, the "Roman" troops were in fact Celts from Mavilly in Gaulish France, and the Romano-Celtic altar to the goddess Verbeia found there is clearly related to those found in the Mavilly region of Gaul, rather than to any other deity worshipped by the locals of West Yorkshire.

The "Roman" soldiers stationed at Cramond were there from approximately 140AD to the third or fourth centuries, and as Jessie Mothersole has commented:

It is surely noteworthy that from an inscribed altar dedicated to Jupiter found within the Roman fort at Cramond, we know that it was staffed at one time by the Fifth Cohort of Gauls (Cohors V Gallorum):
Could it be that, given the likely Romano-Celtic provenance of Eagle Rock, it was indeed connected with the nearby Roman fort, and like the Ilkley Verbeia, was erected by a group of Continental Celts importing their own highly Romanised iconography from Gaul?
Footnotes
[1] J.P. Wood, The Antient and Modern State of the Parish of Cramond (1794).
[2] Rev. W.L. Colvin, "Parish of Cramond", in The New Statistical Account of Scotland (1845), Vol. 1, p.592.
[3] Anne Ross, Pagan Celtic Britain: Studies in Iconography and Tradition (London: Cardinal, 1974), p.188.
[4] Jessie Mothersole, In Roman Scotland (London: Lane & Bodley Head, 1927), p.36.
